Using the average correlation to evaluate Cronbach's alpha

Cronbach's alpha is used as a means of testing the reliability between a set of items and is related to the (arithmetic) average of the absolute values of the Pearson (off-diagonal) correlations. For n variables in a n by n correlation matrix there will be n(n-1)/2 distinct off-diagonal correlations ie taking either the upper or lower triangle of correlations.

As an illustration Kenny (1979, p.125) gives an example for four correlations based upon 4 variables relating to judgments of persons in a mock trial. The six distinct correlations are given in bold in the table below.

Cronbach's alpha (Kenny, 1979 p.132-133) is equal to

$$\frac{n\bar{r}}{1 + (n-1)\bar{r}}$$ where $$\bar{r}$$ is the arithmetic absolute value of the correlations and n is the number of variables.

In the above example, n=4 and there are (4x3/2 =) 6 distinct off-diagonal Pearson correlations so the average r = (0.412+0.629+0.403+0.585+0.270+0.500)/6 = 0.4665 so Cronbach's alpha = [4 (0.4665)] / [1 + 3(0.4665)] = 0.778.

Raykov (1998) has demonstrated that Cronbach's alpha may over- or under-estimate scale reliability. Underestimation is common. For this reason, rho is now preferred and may lead to higher estimates of true reliability. See Raykov (1997), which lists EQS and LISREL code for computing composite reliability. EQS is stand alone software and available at CBSU, as is LISREL.

References

Kenny DA (1979) Correlation and Causality. Wiley:New York. Raykov T (1997). Estimation of composite reliability for congeneric measures. Applied Psychological Measurement, 21, 173-184